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Mirrors > Home > MPE Home > Th. List > Mathboxes > naryrcl | Structured version Visualization version GIF version |
Description: Reverse closure for n-ary (endo)functions. (Contributed by AV, 14-May-2024.) |
Ref | Expression |
---|---|
naryfval.i | ⊢ 𝐼 = (0..^𝑁) |
Ref | Expression |
---|---|
naryrcl | ⊢ (𝐹 ∈ (𝑁-aryF 𝑋) → (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-naryf 45391 | . 2 ⊢ -aryF = (𝑥 ∈ ℕ0, 𝑛 ∈ V ↦ (𝑛 ↑m (𝑛 ↑m (0..^𝑥)))) | |
2 | 1 | elmpocl 7376 | 1 ⊢ (𝐹 ∈ (𝑁-aryF 𝑋) → (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 = wceq 1539 ∈ wcel 2112 Vcvv 3407 (class class class)co 7143 ↑m cmap 8409 0cc0 10560 ℕ0cn0 11919 ..^cfzo 13067 -aryF cnaryf 45390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pr 5291 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-ral 3073 df-rex 3074 df-v 3409 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-br 5026 df-opab 5088 df-xp 5523 df-dm 5527 df-iota 6287 df-fv 6336 df-ov 7146 df-oprab 7147 df-mpo 7148 df-naryf 45391 |
This theorem is referenced by: naryfvalelfv 45396 0aryfvalelfv 45399 fv1arycl 45401 1arymaptfv 45404 fv2arycl 45412 2arymaptfv 45415 |
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